New refinements of fractional Hermite–Hadamard inequality

  • Muhammad Uzair AwanEmail author
  • Muhammad Aslam Noor
  • Ting-Song Du
  • Khalida Inayat Noor
Original Paper


Some new refinements of Hermite–Hadamard type inequalities are obtained. These results involve some different types of fractional integrals. Special cases which are naturally included in the main results of the paper are also discussed.


Convex Strongly convex Hermite–Hadamard Fractional Katugampola fractional integrals Inequalities 

Mathematics Subject Classification

26D15 26A51 26A33 


  1. 1.
    Chen, H., Katugampola, U.N.: Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals. J. Math. Anal. Appl. 446, 1274–1291 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cristescu, G., Lupşa, L.: Non-connected convexities and applications. Kluwer Academic Publishers, Dordrecht (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cristescu, G., Noor, M.A., Awan, M.U.: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpath. J. Math. 31(2), 173–180 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11(5), 91–95 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dragomir, S.S., Pearce, C.E.M.: Selected topics on Hermite-Hadamard inequalities and applications. Victoria University, Footscray (2000)Google Scholar
  6. 6.
    İşcan, I., Wu, S.: Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals. Appl. Math. Comput. 238, 237–244 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Jleli, M., O’Regan, D., Samet, B.: On Hermite-Hadamard type inequalities via generalized fractional integrals. Turkish J. Math. 40, 1221–1230 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  10. 10.
    Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequ. Math. 80, 193–199 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Noor, M.A., Noor, K.I., Awan, M.U., Khan, S.: Fractional Hermite-Hadamard inequalities for some new classes of Godunova-Levin functions. Appl. Math. Inf. Sci. 8(6), 2865–2872 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nikodem, K., Páles, Z.: Characterizations of inner product spaces by strongly convex functions. Banach J. Math. Anal. 5(1), 83–87 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966)Google Scholar
  14. 14.
    Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Modell. 57(9), 2403–2407 (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Set, E.: New inequalities of Ostrowski type for mappings whose derivatives are \(s\)-convex in the second sense via fractional integrals. Comput. Math. Appl. 63(7), 1147–1154 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  • Muhammad Uzair Awan
    • 1
    Email author
  • Muhammad Aslam Noor
    • 2
    • 3
  • Ting-Song Du
    • 4
  • Khalida Inayat Noor
    • 2
  1. 1.Government College UniversityFaisalabadPakistan
  2. 2.Mathematics DepartmentCOMSATS Institute of Information TechnologyIslamabadPakistan
  3. 3.Department of MathematicsKing Saud UniversityRiyadhSaudi Arabia
  4. 4.Department of Mathematics, College of ScienceChina Three Gorges UniversityYichangPeople’s Republic of China

Personalised recommendations